Skip to main content

polynomials - How to set/adjust Precision for an iterative calculation?


How should I restructure this code? I generate a high-order polynomial poly with integer coefficients, then find roots and divide them out of poly one at a time. I want FindRoot[] to use a very high precision on the first pass but then just use the precision of the inputs afterward.



z[n_, c_] := If[n > 0, z[n - 1, c]^2 + c, c];
poly = PolynomialQuotient[z[10, c] - z[6, c], 1 + c^2, c];
rts = {};
Do[Print[Precision[poly]];
 aa = FindRoot[poly, {c, I}, WorkingPrecision -> Min[500, Floor[Precision[poly]]]];
 AppendTo[rts, c /. aa];
 poly = PolynomialQuotient[poly, ((z - c)*(z - Conjugate[c]) /. aa) /. z -> c, c], {j, 1, 5}]

(* [Infinity] 319.649 135.906 0.00598703 *)


I'm losing so much precision on each pass that I can only get a few roots. I'm trying to get the roots closest to i without having to find all the roots. I encountered this precision problem while trying to fix this other problem. I'm also not clear on WHY the polynomial division loses precision so quickly.




Answer



Precision is preserved, and error messages are eliminated by replacing the second instance of PolynomialQuotient by a simple divide.


poly = poly/ (((z - c)*(z - Conjugate[c]) /. aa) /. z -> c)

The only apparent difference is that PolynomialQuotient discards any remainder, and there is a remainder unless c is exact. In some way that I do not understand, discarding the remainder must reduce the precision of poly. So, this may not be a particularly satisfying answer, but it does produce accurate roots for a WorkingPrecision as low as Min[190, Floor[Precision[poly]]]. Replace 190 by 187, however, and the precision of poly gradually decreases to 185, whereupon the Jacobian becomes singular. For completeness, rts[[5]] for 190 is


(* -0.01660571703737496762392836921351877966202681662442570475568531876233767935059224313655985208958046768635767201058325661435694155300994327878649425388424142524075655339495830479365786600752927
 + 1.006001836522824948881217257805018657146542248017733702434346228688166066459126472333560301933200993465308529209500537579261396301606919993043401875320490162101139793328283528897296023734789 I *)

Comments

Post a Comment

Popular posts from this blog

functions - Get leading series expansion term?

Given a function f[x] , I would like to have a function leadingSeries that returns just the leading term in the series around x=0 . For example: leadingSeries[(1/x + 2)/(4 + 1/x^2 + x)] x and leadingSeries[(1/x + 2 + (1 - 1/x^3)/4)/(4 + x)] -(1/(16 x^3)) Is there such a function in Mathematica? Or maybe one can implement it efficiently? EDIT I finally went with the following implementation, based on Carl Woll 's answer: lds[ex_,x_]:=( (ex/.x->(x+O[x]^2))/.SeriesData[U_,Z_,L_List,Mi_,Ma_,De_]:>SeriesData[U,Z,{L[[1]]},Mi,Mi+1,De]//Quiet//Normal) The advantage is, that this one also properly works with functions whose leading term is a constant: lds[Exp[x],x] 1 Answer Update 1 Updated to eliminate SeriesData and to not return additional terms Perhaps you could use: leadingSeries[expr_, x_] := Normal[expr /. x->(x+O[x]^2) /. a_List :> Take[a, 1]] Then for your examples: leadingSeries[(1/x + 2)/(4 + 1/x^2 + x), x] leadingSeries[Exp[x], x] leadingSeries[(1/x + 2 + (1 - 1/x...
Post a Comment
Read more

mathematical optimization - Minimizing using indices, error: Part::pkspec1: The expression cannot be used as a part specification

I want to use Minimize where the variables to minimize are indices pointing into an array. Here a MWE that hopefully shows what my problem is. vars = u@# & /@ Range[3]; cons = Flatten@ { Table[(u[j] != #) & /@ vars[[j + 1 ;; -1]], {j, 1, 3 - 1}], 1 vec1 = {1, 2, 3}; vec2 = {1, 2, 3}; Minimize[{Total@((vec1[[#]] - vec2[[u[#]]])^2 & /@ Range[1, 3]), cons}, vars, Integers] The error I get: Part::pkspec1: The expression u[1] cannot be used as a part specification. >> Answer Ok, it seems that one can get around Mathematica trying to evaluate vec2[[u[1]]] too early by using the function Indexed[vec2,u[1]] . The working MWE would then look like the following: vars = u@# & /@ Range[3]; cons = Flatten@{ Table[(u[j] != #) & /@ vars[[j + 1 ;; -1]], {j, 1, 3 - 1}], 1 vec1 = {1, 2, 3}; vec2 = {1, 2, 3}; NMinimize[ {Total@((vec1[[#]] - Indexed[vec2, u[#]])^2 & /@ R...
Post a Comment
Read more

What is and isn't a valid variable specification for Manipulate?

I have an expression whose terms have arguments (representing subscripts), like this: myExpr = A[0] + V[1,T] I would like to put it inside a Manipulate to see its value as I move around the parameters. (The goal is eventually to plot it wrt one of the variables inside.) However, Mathematica complains when I set V[1,T] as a manipulated variable: Manipulate[Evaluate[myExpr], {A[0], 0, 1}, {V[1, T], 0, 1}] (*Manipulate::vsform: Manipulate argument {V[1,T],0,1} does not have the correct form for a variable specification. >> *) As a workaround, if I get rid of the symbol T inside the argument, it works fine: Manipulate[ Evaluate[myExpr /. T -> 15], {A[0], 0, 1}, {V[1, 15], 0, 1}] Why this behavior? Can anyone point me to the documentation that says what counts as a valid variable? And is there a way to get Manpiulate to accept an expression with a symbolic argument as a variable? Investigations I've done so far: I tried using variableQ from this answer , but it says V[1...
Post a Comment
Read more